Solving Polynomial Equations

Summary:
In algebra you spend lots of time solving polynomial
equations or factoring polynomials (which is the same thing).
It would be easy to get lost in all the techniques, but this paper
ties them all together in a coherent whole.

The Master Plan

Factor = Root

Make sure you aren’t confused by the terminology. All of these are
the same:

Solving a polynomial equation p(x) = 0

Finding roots of a polynomial equation p(x) = 0

Finding zeroes of a polynomial function p(x)

Factoring a polynomial function p(x)

There’s a factor for every root, and vice versa.
(x−r) is a factor if and only if r is a root. This is the
Factor Theorem: finding the roots or finding the factors is
essentially the same thing. (The main difference is how you treat a
constant factor.)

Exact or Approximate?

Most often when we talk about solving an equation or factoring
a polynomial, we mean an exact (or analytic) solution. The
other type, approximate (or numeric) solution, is
always possible and sometimes is the only possibility.

When you can find it, an exact solution is better.
You can always find a numerical approximation to an exact solution,
but going the other way is much more difficult. This page spends most
of its time on methods for exact solutions, but also tells you what to
do when analytic methods fail.

Step by Step

How do you find the factors or zeroes of a polynomial (or the roots
of a polynomial equation)? Basically, you whittle. Every time
you chip a factor or root off the polynomial, you’re left with a
polynomial that is one degree simpler. Use that new reduced
polynomial to find the remaining factors or roots.

At any stage in the procedure, if you get to a
cubic or quartic equation (degree 3 or 4), you have a choice
of continuing with factoring or using the
cubic or quartic formulas. These formulas are a lot
of work, so most people prefer to keep factoring.

Follow this procedure step by step:

If solving an equation, put it in standard form with 0
on one side and simplify.
[ details ]

If you’re down to a linear or quadratic equation
(degree 1 or 2), solve by inspection or the quadratic formula.
[ details ]
Then go to step 7.

Find one rational factor or root. This is the hard part,
but there are lots of techniques to help you.
[ details ]
If you can find a factor or root, continue with step 5 below; if
you can’t, go to step 6.

Divide by your factor. This leaves you with a new
reduced polynomial whose degree is 1 less.
[ details ]
For the rest of the problem, you’ll work with the reduced
polynomial and not the original. Continue at step 3.

If you can’t find a factor or root, turn to
numerical methods.
[ details ]
Then go to step 7.

If this was an equation to solve, write down the roots.
If it was a polynomial to factor, write it in factored form,
including any constant factors you took out in step 1.

This is an example of an algorithm, a set of steps
that will lead to a desired result in a finite number of operations.
It’s an iterative strategy, because the middle steps are
repeated as long as necessary.

Cubic and Quartic Formulas

The methods given here—find a rational root and
use synthetic division—are the easiest.
But if you can’t find a rational root, there are special methods for
cubic equations (degree 3) and
quartic equations (degree 4), both at Mathworld.
An alternative approach is provided by
Dick Nickalls in PDF for
cubic
and
quartic
equations.

Step 1. Standard Form and Simplify

This is an easy step—easy to overlook, unfortunately.
If you have a polynomial equation, put all terms on one side
and 0 on the other.
And whether it’s a factoring problem or an equation to solve, put
your polynomial in standard form, from highest to lowest power.

For instance, you cannot solve this equation in this form:

x³ + 6x² + 12x = −8

You must change it to this form:

x³ + 6x² + 12x + 8 = 0

Also make sure you have simplified, by factoring out any
common factors. This may include factoring out a −1
so that the highest power has a positive coefficient. Example: to factor

7 − 6x − 15x² − 2x³

begin by putting it in standard form:

−2x&sup3 − 15x² − 6x + 7

and then factor out the −1

−(2x³ + 15x² + 6x − 7)
or
(−1)(2x³ + 15x² + 6x − 7)

If you’re solving an equation, you can throw away any
common constant factor. But if you’re factoring a polynomial, you must
keep the common factor.

Example: To solve
8x² + 16x + 8 = 0, you can
divide left and right by the common factor 8. The equation
x² + 2x + 1 = 0
has the same roots as the original equation.

Example: To factor
8x² + 16x + 8 , you recognize the
common factor of 8 and rewrite the polynomial as
8(x² + 2x + 1), which is
identical to the original polynomial.
(While it’s true that you
will focus your further factoring efforts on
x² + 2x + 1, it would be an error
to write that the original polynomial equals
x² + 2x + 1.)

Your “common factor” may be a
fraction, because you must factor out any fractions so that the
polynomial has integer coefficients.

Example: To solve
(1/3)x³ + (3/4)x² − (1/2)x + 5/6 = 0,
you recognize the common factor of 1/12 and divide both sides by 1/12.
This is exactly the same as recognizing and multiplying by the
lowest common denominator of 12. Either way, you get
4x³ + 9x² − 6x + 10 = 0,
which has the same roots as the original equation.

Example: To factor
(1/3)x³ + (3/4)x² − (1/2)x + 5/6,
you recognize the common factor of 1/12 (or the lowest common
denominator of 12) and factor out 1/12. You get
(1/12)(4x³ + 9x² − 6x + 10),
which is identical to the original polynomial.

Step 2. How Many Roots?

A polynomial of degree n will have n roots, some of which may be
multiple roots.

How do you know this is true? The
Fundamental Theorem of Algebra tells you that the polynomial
has at least one root. The Factor Theorem tells you that if r
is a root then (x−r) is a factor. But if you divide a polynomial
of degree n by a factor (x−r), whose degree is 1, you get a
polynomial of degree n−1. Repeatedly applying the Fundamental
Theorem and Factor Theorem gives you n roots and n factors.

Descartes’ Rule of Signs

Descartes’ Rule of Signs can tell you how many positive and
how many negative real zeroes the polynomial has. This is a
big labor-saving device, especially when you’re deciding which
possible rational roots to pursue.

To apply Descartes’ Rule of Signs, you need to understand the term
variation in sign. When the polynomial is arranged in
standard form, a variation in
sign occurs when the sign of a coefficient is different from the sign
of the preceding coefficient. (A zero coefficient is ignored.) For
example,

p(x) = x5 − 2x3 + 2x2 − 3x + 12

has four variations in sign.

Descartes’ Rule of Signs:

The number of positive roots of p(x)=0 is either equal to the
number of variations in sign of p(x), or less than that by an even
number.

The number of negative roots of p(x)=0 is either equal to the
number of variations in sign of p(−x), or less than that by an even
number.

Example: Consider p(x) above. Since it has four variations
in sign, there must be either four positive roots, two positive roots,
or no positive roots.

Now form p(−x), by replacing x with (−x) in the
above:

p(−x) =(−x)5 − 2(−x)3 + 2(−x)2 − 3(−x) + 12

p(−x) = −x5 + 2x3 + 2x2 + 3x + 12

p(−x) has one variation in sign, and therefore the
original p(x) has one negative
root. Since you know that p(x) must have a negative root, but it may
or may not have any positive roots, you would look first for negative
roots.

p(x) is a fifth−degree polynomial, and therefore it must have five zeros.
Since x is not a factor, you know that x=0 is not a
zero of the polynomial. (For a polynomial with real coefficients, like
this one, complex roots occur in pairs.)
Therefore there are three possibilities:

number of zeroesthat are

positive

negative

complexnot real

first possibility

4

1

0

second possibility

2

1

2

third possibility

0

1

4

Complex Roots

If a polynomial has real coefficients, then either all
roots are real or there are an
even number of non-real complex roots, in conjugate pairs.

For example, if 5+2i is a zero of a polynomial with real
coefficients, then 5−2i must also be a zero of that polynomial.
It is equally true that if (x−5−2i) is a factor then
(x−5+2i) is also a factor.

Why is this true? Because when you have a factor with an imaginary
part and multiply it by its complex conjugate you get a real
result:

(x−5−2i)(x−5+2i) =
x²−10x+25−4i² =
x²−10x+29

If (x−5−2i) was a factor but
(x−5+2i) was not, then the polynomial would end up with
imaginaries in its coefficients, no matter what the other factors
might be. If the polynomial has only real
coefficients, then any complex roots must occur in conjugate pairs.

Irrational Roots

For similar reasons, if the polynomial has
rational coefficients then the irrational roots involving
square
roots occur (if at all) in conjugate pairs.
If (x−2+√3) is a factor of a polynomial with rational
coefficients, then (x−2−√3) must also be a
factor. (To see why, remember how you rationalize a binomial
denominator; or just check what happens when you multiply those two
factors.)

As Jeff Beckman pointed out (20 June 2006), this
is emphatically not true for odd roots. For instance,
x³−2 = 0 has three roots, 2^(1/3) and two
complex roots.

It’s an interesting problem whether irrationals
involving even roots of order ≥4 must also occur in conjugate
pairs. I don’t have an immediate answer. I’m working on a
proof, as I have time.

Multiple Roots

When a given factor (x−r) occurs m times in a polynomial, r is
called a multiple root or a root of multiplicity m

If the multiplicity m is an even number, the graph touches the
x axis at x=r but does not cross it.

If the multiplicity m is an odd number, the graph crosses the
x axis at x=r. If the multiplicity is 3, 5, 7, and so on, the graph is
horizontal at the point where it crosses the axis.

Examples: Compare these two polynomials and their graphs:

f(x) = (x−1)(x−4)2 = x3 − 9x2 + 24x − 16

g(x) = (x−1)3(x−4)2 = x5 − 11x4 + 43x3 − 73x2 + 56x − 16

These polynomials have the same zeroes, but the root 1 occurs
with different multiplicities. Look at the graphs:

Both polynomials have zeroes at 1 and 4 only. f(x) has degree 3,
which means three roots. You see from the factors that 1 is a root of
multiplicity 1 and 4 is a root of multiplicity 2. Therefore the graph
crosses the axis at x=1 (but is not horizontal there) and touches at
x=4 without crossing.

By contrast, g(x) has degree 5. (g(x) = f(x) times
(x-1)2.) Of the five roots, 1 occurs with
multiplicity 3: the graph crosses the axis at x=1 and is horizontal
there; 4 occurs with multiplicity 2, and the graph touches the
axis at x=4 without crossing.

Step 3. Quadratic Factors

When you have quadratic factors (Ax²+Bx+C), it may or may not
be possible to factor them further.

Sometimes you can just see the factors, as with
x²−x−6 = (x+2)(x−3).
Other times it’s not so obvious whether the
quadratic can be factored. That’s when the quadratic formula
(shown at right) is your friend.

For example, suppose you have a factor of
12x²−x−35. Can that be factored further? By trial and
error you’d have to try a lot of combinations! Instead, use the fact
that factors correspond to roots, and apply the formula to
find the roots of 12x²−x−35 = 0, like this:

x = [ −(−1) ± √[1 − 4(12)(−35)] ] / 2(12)

x = [ 1 ± √1681 ] / 24

√1681 = 41, and therefore

x = [ 1 ± 41 ] / 24

x = 42/24 or −40/24

x = 7/4 or −5/3

If 7/4 and −5/3 are roots, then (x−7/4) and (x+5/3)
are factors. Therefore

12x²−x−35 = (4x−7)(3x+5)

What about x²−5x+7? This one looks like it’s prime,
but how can you be sure? Again, apply the formula:

x = [ −(−5) ± √[25 − 4(1)(7)] ] / 2(1)

x = [ 5 ± √(−3) ] / 2

What you do with that depends on the original problem. If it
was to factor over the reals, then x²−5x+7 is prime. But if
that factor was part of an equation and you were supposed to find all
complex roots, you have two of them:

x = 5/2 + ((√3)/2)i, x = 5/2 − ((√3)/2)i

Since the original equation had real coefficients, these
complex roots occur in a conjugate pair.

Step 4. Find One Factor or Root

This step is the heart of factoring a polynomial or solving a
polynomial equation. There are a lot of techniques that can help you
to find a factor.

Sometimes you can find factors by inspection (see the first two
sections that follow). This provides a great shortcut, so check for
easy factors before starting more strenuous
methods.

Monomial Factors

Always start by looking for any monomial factors you can see. For instance,
if your function is

f(x) = 4x6 + 12x5 + 12x4 + 4x3

you should immediately factor it as

f(x) = 4x3(x3 + 3x2 + 3x + 1)

Getting the 4 out of there simplifies the remaining numbers, the
x3 gives you a root of x = 0 (with multiplicity
3), and now you have only a cubic polynomial (degree 3) instead of a
sextic (degree 6). In fact, you should now recognize that cubic as a
special product, the perfect cube
(x+1)3.

When you factor out a common variable factor, be sure you
remember it at the end when you’re listing the factor or roots.
x³+3x²+3x+1 = 0 has certain roots, but
x³(x³+3x²+3x+1) = 0 has those same roots and
also a root at x=0.

Special Products

Be alert for applications of the Special Products.
If you can apply them, your task becomes much easier. The Special
Products are

Rational Roots

The answer is the Rational Root Test.
It can show you some candidate roots
when you don’t see how to factor the polynomial, as follows.

Consider a polynomial in standard form, written from highest degree
to lowest and with only integer coefficients:

f(x) = anxn + ... + ao

The Rational Root Theorem tells you that if the
polynomial has a rational zero
then it must be a fraction p/q,
where p is a factor of the trailing constant ao and
q is a factor of the leading coefficient an.

Example:

p(x) = 2x4 − 11x3 − 6x2 + 64x + 32

The factors of the leading coefficient (2) are 2 and 1. The
factors of the constant term (32) are 1, 2, 4, 8, 16, and 32.
Therefore the possible rational zeroes are ±1, 2, 4, 8, 16, or
32 divided by 2 or 1:

±1/2, 1/1, 2/2, 2/1, 4/2, 4/1, 8/2, 8/1, 16/2, 16/1, 32/2, 32/1

reduced: ± ½, 1, 2, 4, 8, 16, 32

What do we mean by saying this is a list of all the
possible rational roots? We mean that no other rational number,
like ¼ or 32/7, can be a root of p(x) = 0.

Caution: Don’t make the Rational Root Test out to be
more than it is. It doesn’t say those rational numbers are roots, just
that no other rational numbers can be roots. And it doesn’t tell
you anything about whether some irrational or even complex roots
exist. The Rational Root Test is only a starting point.

Suppose you have a polynomial with non-integer coefficients.
Are you stuck? No, you can factor out the least common
denominator (LCD) and get a polynomial with integer coefficients that
way. Example:

(1/2)x³ − (3/2)x² + (2/3)x − 1/2

The LCD is 1/6. Factoring out 1/6 gives the polynomial

(1/6)(3x³ − 9x² + 4x − 3)

The two forms are equivalent, and therefore they have the same
roots. But you can’t apply the Rational Root Test to the first form,
only to the second. The test tells you that the only possible rational
roots are ±1/3, 1, 3.

Once you’ve identified the possible rational zeroes, how
can you screen them? The brute-force method would be to take each
possible value and substitute it for x in the polynomial: if the
result is zero then that number is a root. But there’s a better
way.

Use Synthetic Division to see if each
candidate makes the polynomial equal zero. This is better for three
reasons. First, it’s computationally easier, because you don’t have to
compute higher powers of numbers. Second, at the same time it tells
you whether a given number is a root, it produces the
reduced polynomial that you’ll use to find the remaining
roots. Finally, the results of synthetic division may give you an
upper or lower bound even if the number you’re
testing turns out not to be a root.

Sometimes Descartes’ Rule of Signs can
help you screen the possible rational roots further. For example, the
Rational Root Test tells you that if

q(x) = 2x4 + 13x3 + 20x2 + 28x + 8

has any rational roots, they must come from the list
±½, 1, 2, 4, 8. But don’t just start off substituting or
synthetic dividing. Since there are no sign changes, there are no
positive roots. Are there any negative roots?

q(−x) = 2x4 − 13x3 + 20x2 − 28x + 8

has four sign changes. Therefore there could be as many as four
negative roots. (There could also be two negative roots, or none.)
There’s no guarantee that any of the roots are rational, but any root
that is rational must come from the list −½, −1,
−2, −4, −8.

(If you have a
graphing calculator, you can pre-screen the rational roots by graphing
the polynomial and seeing where it seems to cross the x axis. But you
still need to verify the root algebraically, to see that f(x) is
exactly 0 there, not just nearly 0.)

Remember, the Rational Root Test guarantees to find all
rational roots. But it will completely miss real roots that are not
rational, like the roots of x²−2=0, which are
±√2, or the roots of x²+4=0, which are
±2i.

Finally, remember that the Rational Root Test works only if all
coefficients are integers. Look again at this function, which is
graphed at right:

p(x) = 2x4 − 11x3 − 6x2 + 64x + 32

The Rational Root Theorem tells you that the only possible rational
zeroes are ±½, 1, 2, 4, 8, 16, 32. But suppose you
factor out the 2 (as I once did in class), writing the equivalent
function

p(x) = 2(x4 − (11/2)x3 − 3x2 + 32x + 16)

This function is the same as the earlier one, but you can no
longer apply the Rational Root Test because the coefficients are not
integers. In fact −½ is a zero of p(x), but it did not
show up when I (illegally) applied the Rational Root Test to the
second form. My mistake was forgetting that the Rational Root Theorem applies
only when all coefficients of the polynomial are
integers.

Graphical Clues

By graphing the function—either by hand or with a graphing
calculator—you can get a sense of where the roots are,
approximately, and how many real roots exist.

Example: If the Rational Root Test
tells you that ±2 are possible rational roots, you can look at
the graph to see if it crosses (or touches) the x axis at 2 or
−2. If so, use synthetic division to
verify that the suspected root actually is a root. Yes, you always
need to check—from the graph you can never be sure
whether the intercept is at your possible rational root or
just near it.

Boundaries on Roots

Some techniques don’t tell you the specific value of a root, but
rather that a root exists between two values or that all roots are
less than a certain number of greater than a certain number. This
helps narrow down your search.

Intermediate Value Theorem

This theorem tells you that if the graph of a polynomial is above
the x axis for one value of x and below the x axis for
another value of x, it must cross the x axis somewhere between.
(If you can graph the function, the crossings
will usually be obvious.)

Example:

p(x) = 3x³ + 4x² − 20x −32

The rational roots
(if any) must come from the list
±1/3, 2/3, 1, 4/3, 2, 8/3, 4, 16/3, 8, 32/3, 16, 32.
Naturally you’ll look at the integers first, because the arithmetic is
easier. Trying synthetic division, you
find p(1) = −45, p(2) = −22, and
p(4) = 144. Since p(2) and p(4) have opposite signs, you
know that the graph crosses the axis between x=2 and x=4, so there is
at least one root between those numbers. In other words, either 8/3 is
a root, or there root(s) between 2 and 4 are irrational. (In fact,
synthetic division reveals that 8/3 is a root.)

The Intermediate Value Theorem can tell you where there is a
root, but it can’t tell you where there is no root. For example,
consider

q(x) = 4x² − 16x + 15

q(1) and q(3) are both positive, but that doesn’t tell you
whether the graph might touch or cross the axis between. (It actually
crosses the axis twice, at x = 3/2 and
x = 5/2.)

Upper and Lower Bounds

One side effect of synthetic division is
that even if the number you’re testing turns out as not a root, it may
tell you that all the roots are smaller or larger than that
number:

If you do synthetic division by a positive number a, and every
number in the bottom row is positive or zero, then a is an
upper bound for the roots, meaning that all the real roots
are ≤ a.

If you do synthetic division by a
negative number b, and the numbers in the bottom row alternate
sign, then b is a lower bound for the roots, meaning that all
the real roots are ≥ b.

What if the bottom row contains zeroes? A more complete
statement is that alternating nonnegative and nonpositive signs,
after synthetic division by a negative number, show a lower bound on
the root. The next two examples clarify that.

(By the way, the rule for lower bounds follows
from the rule for upper bounds.
Lower limits on roots of p(x) equal upper limits on
roots of p(−x), and dividing by (−x+r) is the same as
dividing by −(x−r).)

Example:

q(x) = x3 + 2x2 − 3x − 4

Using the Rational Root
Test, you identify the only possible rational roots as
±4, ±2, and ±1. You decide to try −2 as a
possible root, and you test it with synthetic division:

-2 | 1 2 -3 -4
| -2 0 6
|------------------
1 0 -3 2

−2 is not a root of the equation f(x)=0.
The third row shows alternating signs, and you were dividing by a
negative number; however, that zero mucks things up.
Recall that you have a lower bound only if the signs in the bottom row
alternate nonpositive and nonnegative. The 1 is positive
(nonnegative), and the 0 can count as nonpositive, but the
−3 doesn’t qualify as nonnegative. The alternation is
broken, and you do not know whether there are roots
smaller than −2. (In fact, graphical or
numerical methods would show a root around −2.5.)
Therefore you need to try the lower possible rational root, −4:

-4 | 1 2 -3 -4
| -4 8 -20
|------------------
1 -2 5 -24

Here the signs do alternate; therefore you know there are no
roots below −4. (The remainder −24 shows you that
−4 itself isn’t a root.)

Here’s another example:

r(x) = x³ + 3x² − 3

The Rational Root Test tells you
that the possible rational roots are ±1 and ±3. With synthetic
division for −3:

-3 | 1 3 0 -3
| -3 0 0
|------------------
1 0 0 -3

−3 is not a root, but the signs do alternate here, since
the first 0 counts as nonpositive and the second as nonnegative.
Therefore −3 is a lower bound to the roots, meaning that
the equation has no real roots lower than −3.

Coefficients and Roots

There is an interesting relationship between the coefficients of a
polynomial and its zeroes. I mention it last because it is more suited
to forming a polynomial that has zeroes with desired properties,
rather than finding zeroes of an existing polynomial. However, if you
know all the roots of a polynomial but one or two, you can easily use this
technique to find the remaining root.

Consider the polynomial

f(x) = anxn + an−1xn−1 + an−2xn−2 + ... + a2x2 + a1x + ao

The following relationships exist:

−an−1÷an = sum of all the roots

+an−2÷an = sum of the products of roots
taken two at a time

−an−3÷an = sum of the products of roots
taken three at a time

and so forth, until

(−1)na0÷an = product of all the roots

Example: f(x) = x3 − 6x2 −
7x − 8 has degree 3, and therefore at most three real zeroes. If
we write the real zeroes as r1, r2,
r3, then the sum of the roots is
r1+r2+r3 = −(−6) = 6; the
sum of the products of roots taken two at a time is
r1r2+r1r3+r2r3 =
−7, and the product of the roots is
r1r2r3 =
(−1)3(−8) = 8.

Example: Given that the polynomial

g(x) = x5 − 11x4 + 43x3 − 73x2 + 56x − 16

has a triple root at x=1, find the other two roots.

Solution: Let the other two roots be c and d. Then you know that
the sum of the all roots is 1+1+1+c+d = −(−11) = 11, or
c+d = 8. You also know that the product of all the roots is
1×1×1×cd = (−1)5(−16) = 16, or
cd = 16. c+d = 8, cd = 16; therefore c = d =
4, so the remaining roots are a double root at x=4.

More Coefficients and Roots

There are several further theorems about the relationship
between coefficients and roots.
Wikipedia’s article
Properties of Polynomial Roots
gives a good though somewhat terse summary.

Step 5. Divide by Your Factor

Remember that r is a root if and only if x−r is a factor;
this is the Factor Theorem. So if you want
to check whether r is a root, you can divide the polynomial by
x−r and see whether it comes out even (remainder of 0).
Elizabeth Stapel has a nice
example of dividing polynomials by long division.

Synthetic division also has some side benefits. If your suspected
root actually is a root, synthetic division gives you the
reduced polynomial. And sometimes you also luck out and
synthetic division shows you an upper or lower
bound on the roots.

You can use synthetic division when you’re dividing by a
binomial of the form x−r for a constant r. If you’re dividing by
x−3, you’re testing whether 3 is a root and you synthetic divide
by 3 (not −3). If you’re dividing by x+11, you’re testing
whether −11 is a root and you synthetic divide by −11 (not
11). −11 (not 11).

Example:

p(x) = 4x4 − 35x2 − 9

You suspect that x−3 might be a factor, and you test it by
synthetic division, like this:

3 | 4 0 -35 0 -9
| 12 36 3 9
|--------------------
4 12 1 3 0

Since the remainder is 0, you know that 3 is a root of
p(x) = 0, and x−3 is a factor of p(x). But you know
more. Since 3 is positive and the bottom row of the synthetic division
is all positive or zero, you know that all the roots of
p(x) = 0 must be ≤ 3. And you also know
that

p(x) = (x−3)(4x3 + 12x2 + x + 3)

4x3 + 12x2 + x + 3
is the reduced polynomial. All of its factors are also
factors of the original p(x), but its degree is one lower, so it’s
easier to work with.

Step 6. Numerical Methods

When your equation has no more rational roots (or your
polynomial has no more rational factors), you can turn to numerical
methods to find the approximate value of irrational roots:

Many graphing calculators have a “Solve” or
“Root” or “Zero” command to help you find
approximate roots. For instance, on the TI-83 or TI-84, you
graph
the function and then select [2nd] [Calc] [zero].

Complete Example

Solve for all complex roots:

4x³ + 15x − 36 = 0

Step 1. The equation is already in standard form, with
only zero on one side, and powers of x from highest to lowest. There
are no common factors.

Step 2. Since the equation has degree 3, there will be 3
roots. There is one variation in sign, and from
Descartes’ Rule of Signs you know there must
be one positive root. Examine the polynomial with −x replacing
x:

−4x³ − 15x − 36

There are no variations in sign, which means there are no
negative roots. The other two roots must therefore be complex
conjugates.

Steps 3 and 4. The possible rational roots are
unfortunately rather numerous: any of 1, 2, 3, 4, 6, 9, 12, 18, 36
divided by any of 4, 2, 1. (Only positive roots are listed because you
have already determined that there are no negative roots for this
equation.) You decide to try 1 first:

1 | 4 0 15 -36
| 4 4 19
|-----------------
4 4 19 -17

1 is not a root, so you test 2:

2 | 4 0 15 -36
| 8 16 62
|-----------------
4 8 31 26

Alas, 2 is not a root either. But notice that
f(1) = −17 and f(2) = 26. They have opposite
signs, which means that the graph crosses the x axis between x=1
and x=2, and a root is between 1 and 2. (In this case it’s the only
root, since you have determined that there is one positive root and
there are no negative roots.)

The only possible rational root between 1 and 2 is 3/2, and
therefore either 3/2 is a root or the root is irrational. You try 3/2
by synthetic division:

3/2 | 4 0 15 -36
| 6 9 36
|-----------------
4 6 24 0

Hooray! 3/2 is a root. The reduced polynomial is
4x² + 6x + 24. In other words,

(4x³ + 15x − 36) ÷
(x−3/2) =
4x² + 6x + 24

The reduced polynomial has degree 2,
so there is no need for more
trial and error, and you continue to step 5.

Step 5. Now you must solve

4x² + 6x + 24 = 0

First divide out the common factor of 2:

2x² + 3x + 12 = 0

It’s no use trying to factor that quadratic, because
you determined using Descartes’ Rule of Signs that there are no more
real roots. So you use the quadratic
formula:

x = [ −3 ± √[9 − 4(2)(12)] ] / 2(2)

x = [ −3 ± √(−87) ] / 4

x = −3/4 ± ((√87)/4)i

Step 6. Remember that you found a root in an
earlier step! The full list of roots is